Tangent

Date: 02/14/2003 at 10:10:10
From: Laura
Subject: Trigonometry
Is the trigonometric ratio tangent related to the tangent of a circle?

Date: 02/14/2003 at 10:21:20
From: Doctor Jerry
Subject: Re: Trigonometry
Hi Laura,
Yes. Draw a coordinate axis - just as when you sketch an (x,y)-plane.
Draw a unit circle, centered at the origin. Draw a tangent to this
circle, upward from B=(1,0). Draw a line L from (0,0) to this tangent
line - do this so that L is in the first quadrant. Let A be the point
where L meets the tangent line. The line L makes an angle theta with
the x-axis. Note that tan(theta) = AB/1 = AB.
- Doctor Jerry, The Math Forum
http://mathforum.org/dr.math/

Date: 02/17/2003 at 10:44:27
From: Laura
Subject: Trigonometry
Having read about the relation between the tangent of a circle and the
tan ratio used in trigonometry, I wondered if the sine and cosine
ratios were also linked to circles? If so, what about cotangent,
secant and cosecant?

Date: 02/17/2003 at 12:10:04
From: Doctor Jerry
Subject: Re: Trigonometry
Hi Laura,
Yes, one can show that sine and cosine have natural interpretations
(see below); maybe one could give interpretations for cotangent,
secant, and cosecant (each of this is the reciprocal of sine, cosine,
or tangent), but I believe the interpretations would be somewhat
artificial and not terribly useful.
For sine and cosine, draw a coordinate axis, just as when you sketch
an (x,y)-plane. Draw a unit circle, centered at the origin. Draw a
line L from (0,0) to the unit circle. It can be in any quadrant. Let
theta be the angle needed to rotate the positive x-axis
counterclockwise until it conincides with L. Let A=(a,b) be the point
where L intersects the unit circle.
It is quite easy to see that a = cos(theta) and b = sin(theta).
- Doctor Jerry, The Math Forum
http://mathforum.org/dr.math/

Date: 02/17/2003 at 13:58:34
From: Laura
Subject: Thank you (Trigonometry)
Thank you very much for your help in answering my questions. Maths is
my favourite subject and I quite often drive my teacher up the wall
with all my "but why?"'s. I have been able to follow your instructions
and prove how it works for myself. I will show my teacher this week.
Thank you.